You're also getting into Mises' "regression theorem" of money (and kudos for that!). It's a worthwhile thought experiment to ask whether bitcoin affirms or challenges that theorem. I think Robert P. Murphy has done some writing on this, but I don't remember on which side of the argument he concluded. I would suggest that money doesn't originate *only* from things with "commodity" or ornamental or ceremonial use, but also from things that have "social" use. In bitcoin's case, the "social" use is as a trustless store of value and means of exchange, with durability (cryptography), portability/transferability (more so than gold), auditability, fixed supply, extreme divisibility (potentially), fungibility, etc.
In our analogy, what "backs" the coat claim check is the coat, which people value. With bitcoin, what "backs" it are simply the pure attributes of money. We've just cut out the middle man.